Formulae sheet - Department of Zoology, UBC

Formulae sheet - Department of Zoology, UBC

Please do not mark on the formula sheets or tables.
Formulae for Basic Statistics
n
ΣY
Pooled variance
df s2 + df 2 s22
2
sp = 1 1
df1 + df 2
i
Y =
i =1
n
Σ(Yi − Y ) 2
s=
n −1
( )
2
s=
Σ Yi − nY 2
n −1
F=
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Standard error of the mean
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s/ n
t=
2
χ test of goodness-of-fit
(Oi − Ei )2
2
χ =∑
Ei
i
Binomial Probability Distribution
! N$ x
N− x
P[x] = #
p (1 − p)
" x%
Normal Probability Distribution
−
1
P[x] =
e
2πσ 2
( x − µ)
2σ
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2
(Y1 − Y2 )
SE Y1 −Y2
2
tα ( 2),df
Y −µ
s/ n
x −µ
Z=
σ
t=
#1 1&
SE Y −Y = s2p % + (
1 2
$ n1 n 2 '
Y −Y
t= 1 2
" s12 s22 % 2
s12 s22
$ + '
+
# n1 n 2 &
n1 n 2 df =
2
" s2 n 2
s22 n 2 ) %
(
)
(
1
1
$
'
+
$ n1 −1
n 2 −1 '
#
&
Mann-Whitney U
2
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n ( n + 1)
U = n1n 2 + 1 1
− R1
2
U # = n1n 2 − U
Z=
Confidence Interval for the variance of a
normal distribution
df s2
df s2
2
≤
σ
≤
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χ α2
χ2 α
,df
s22
s12
1
Confidence Interval for the mean of a
normal distribution
Y ± SEY tα (2),df
2
or
(Y1 − Y2 ) ± SEY −Y
ln(Oˆ R) − Z α SE[ln(Oˆ R)] ≤ ln(OR) ≤ ln(Oˆ R) + Z α SE[ln(Oˆ R)]
ad
Oˆ R =
bc
Poisson Probability Distribution
µ x e− µ
P[x] =
x!
s12
s22
1− ,df
2
2U − n1n 2
n1n 2 ( n1 + n 2 + 1) /3
Agresti-Coull method: p" =
X +2
.
n+4
$
$
p"(1− p") '
p"(1− p") '
&& p" − Z
)) ≤ p ≤ && p" + Z
)
n +€4 (
n + 4 )(
%
%
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Please do not mark on the formula sheets or tables.
Formulae for regression and correlation
(∑ X )(∑Y )
∑(X − X )(Y − Y ) = ∑( XY ) −
n
∑(X
i
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∑(X
i
− X)
r=
∑(X
#
&
% ∑ Yi (
$
'
= ∑ Yi 2 −
n
2
SSresidual + SSregression = SSTotal
SS
MSx = x
DFx
SS
r 2 = regression
SSTotal
MSresidual
SE b =
2
∑ ( Xi − X )
rs = 1 −
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n3 − n
2
MSerror = s pooled
− b∑ ( X i − X )(Yi − Y )
MSresidual =
2
i
ANOVA etc.
MS groups
F=
MS error
MSgroups
i
∑ (Y − Y )
6∑ di2
SSregression = b ∑ ( Xi − X )(Yi − Y )
∑ (Y − Y )
2
1 − r2
SE r =
n−2
" 1+ r $
z = 0.5ln #
1− r %
1
σz =
n−3
2
2
− X)
i
a = Y − bX
− X )(Yi − Y )
i
− X )(Yi − Y )
b=
SSTotal
∑(X
n −2
Y =
∑ s (n −1)
=
2
i
i
N −k
∑ n (Y −Y )
=
i
2
i
k −1
∑ n (Y )
i
i
N
SS groups
b ± tα [2 ],ν SEb
Yˆ ± t
SE ˆ
R2 =
b − β0
t=
SE b
(b − b2 ) − ( β1 − β2 )
t= 1
SEb1− b2
( SS ) + ( SSerror )2
( MSerror ) p = error 1
( DFerror )1 + ( DFerror )2
Kruskal-Wallis
" Ri 2 %
12
H=
∑ n ' − 3( N + 1)
N ( N + 1) $#
i &
α [2],ν
SEb1−b2 =
SS total
Y
( MSerror ) p
+
( MSerror ) p
#
#
2&
2&
%∑( X − X ) ( %∑( X − X ) (
$
'1 $
'2
Tukey-Kramer:
q=
Yi − Y j
SE
&1
#
2
$ + 1 !.
SE = s pooled
$n
!
% i nj "